RELATIVELY MAXIMUM VOLUME RIGIDITY IN ALEXANDROV GEOMETRY

Given compact metric spaces X and Z with Hausdorff dimension n, if there is a distance-nonincreasing onto map f : Z -> X, then the Hausdorff n-volumes satisfy vol(X) <= vol(Z). The relatively maximum volume conjecture says that if X and Z are both Alexandrov spaces and vol(X) = vol(Z) is isometric to a gluing space produced from Z along its boundary partial derivative Z and f is length-preserving. We partially verify this conjecture and give a further classification for compact Alexandrov n-spaces with relatively maximum volume in terms of a fixed radius and space of directions. We also give an elementary proof for a pointed version of the Bishop-Gromov relative volume comparison with rigidity in Alexandrov geometry.